International Journal of Antennas and Propagation

Volume 2015, Article ID 165935, 14 pages

http://dx.doi.org/10.1155/2015/165935

## Multiantenna Spectrum Sensing in the Presence of Multiple Primary Users over Fading and Nonfading Channels

National Institute of Telecommunications (INATEL), P.O. Box 05, 37540-000 Santa Rita do Sapucaí, MG, Brazil

Received 19 February 2015; Accepted 25 April 2015

Academic Editor: Song Guo

Copyright © 2015 Dayan Adionel Guimarães et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The basis of this paper is Wei and Tirkkonen, 2012, in which expressions for the key performance metrics of the sphericity test applied to the multiantenna cooperative spectrum sensing of multiple primary transmitters in cognitive radio networks over nonfading channels are provided. The false alarm and the detection probabilities were derived in Wei and Tirkkonen, 2012, based on approximations obtained by matching the moments of the test statistics to the Beta distribution. In this paper we show that the model adopted in Wei and Tirkkonen, 2012, does not apply directly to fading channels, yet being considerably inaccurate for some system parameters and channel conditions. Nevertheless, we show that the original expressions from Wei and Tirkkonen, 2012, can be simply and accurately applied to a modified model that considers fixed or time-varying channels with any fading statistic. We also analyze the performance of the sphericity test and other competing detectors with a varying number of primary transmitters, considering different situations in terms of the channel gains and channel dynamics. Based on our results, we correct several interpretations from Wei and Tirkkonen, 2012, in what concerns the performance of the detectors, both over a fixed-gain additive white Gaussian noise channel and over a time-varying Rayleigh fading channel.

#### 1. Introduction

The cognitive radio (CR) [1] concept has come as a promising solution for alleviating the problem of spectrum scarcity in wireless communication systems and is one of the key enabling technologies of the fifth-generation (5G) of these systems [2]. In this concept, unused spectrum bands in the primary (incumbent) network can be opportunistically used by secondary CR networks. In order to accomplish this task, a spectrum sensing [3] technique detects unused bands so that the CRs can use them without causing harm interference to the primary users. In order to increase the reliability of the decisions upon the occupancy of a given channel, cooperative spectrum sensing has become the main choice [3].

As pointed out in [4], most of the literature on cooperative spectrum sensing predominately adopt the assumption of a single primary transmitter. However, this assumption may fail in most real networks where the existence of more than one primary transmitter prevails. In [4] the authors give an important contribution to the theoretical analysis of the performance of the spectrum sensing under multiple primary users. Specifically, they consider a multiantenna cooperative spectrum sensing and adopt the covariance-based technique known as sphericity test. Expressions for the false alarm probability and the detection probability were derived in [4] by means of approximations for the distributions of the test statistics under the hypothesis of absence and presence of the primary signals. The approximations were obtained by matching the moments of the test statistics to the Beta distribution. It is claimed in [4] that the derived approximations are easily computable and that they are accurate for the considered sensor sizes , number of samples , number of primary users , and corresponding signal-to-noise ratios , . Empirical results were compared with analytical ones in order to validate their claims. As an incremental result, in [4] the sphericity test detector has been compared with competing detectors in the presence of multiple primary users.

*Opportunities for Extended Results*. In the numerical results’ section of [4], it is stated that the channel between the primary transmitters and the secondary sensors is considered fixed during the sensing interval, which is a commonly adopted and reasonable assumption if the sensing time is considerably smaller than the coherence time of the channel. It is also stated that the channel gains were independently drawn from a complex Gaussian distribution, corresponding to Rayleigh fading. However, in the sequel the authors consider that the channel is the same in all Monte Carlo simulation runs, which contradicts the fading channel assumption. The simulated channel is in fact a fixed gain additive white Gaussian noise (AWGN) channel with configurable SNRs from the primary transmitters to the sensors. This fact has a major impact on the results, as we show later on in this paper.

Moreover, in [4] the channel gains are normalized to have unitary second moment, which further prevents the derived expressions to be used when the channel is in fact time varying. This is because such normalization changes the fading statistics, making them depart from the predefined ones.

Not less important, to apply the expression of the detection probability derived in [4], one must use a covariance matrix that relies on a single realization of the channel gains. When the channel is considered fixed and the same in all sensing intervals (all simulation runs), and the above-mentioned gain normalization is applied, a good agreement is achieved between theoretical and empirical results. This is because the same covariance matrix applies to all simulation runs and, thus, keeps consistence with the theoretical calculations. Nevertheless, it is reasonable to accept that in a fading channel one must not rely on a single channel realization to predict the system performance over the varying channel gains. In fact, if the channel is made variable and no normalization is applied to the channel gains, rare casual agreements are achieved between empirical results and the theoretical results obtained from the expressions in [4].

It can also be identified that some results in [4] are considerably inaccurate for some system parameters different from those originally reported. It was not claimed in [4] that accuracy is achieved for any system parameter, but it was not mentioned either that inaccuracy could result depending on the choice of these parameters. Moreover, in spite of the fact that the authors of [4] have claimed that the derived approximations are easily computable, computation errors may result depending on the parameters chosen and software package used.

In Section 4.2 of [4], the performance of the sphericity test is compared with other competing detectors in terms of receiver operating characteristic (ROC) curves. In Figures 3–5 of that paper, such comparison has been made under the assumption of fixed and normalized channel, using a single channel realization for all results and all simulation runs. However, different channel realizations change the detection probabilities and, as a consequence, modify the corresponding ROC curves. Since different detection techniques can be differently affected by the channel gains, the performance ranking or the performance gaps or both can be modified from a channel realization to another.

We add that the approach adopted in [4], which motivated the present paper, was also adopted in [5]. Thus, most of the above comments also apply to [5].

The sphericity test alone is also considered in [6], where the authors claim that their contribution is the first to address the cooperative spectrum sensing problem in a multiple primary user scenario, considering multipath fading channels and using eigenvalue-based detectors. In [6], the expression for the probability of detection derived in [4] for the AWGN channel is numerically averaged over the probability density function of the Rayleigh fading signal-to-noise ratio (SNR). However, this approach is not correct because the probability of detection does not depend on the SNR in such a direct manner. In fact, it is a function of the determinant of a covariance matrix, which in turn depends on the channel gains from each primary transmitter to each secondary receiver. Thus, the SNR influence is implicit in this channel gains.

*Contributions*. Having highlighted the limitations of the analytical results in [4], we can list the following main contributions of this paper:(i)A thorough analysis of the channel normalization and channel dynamics on the performance of the sphericity test is made, considering the detection of multiple primary transmitters. Specifically, we analyze four possible channel conditions: fixed and normalized gains, fixed and nonnormalized gains, time-varying and normalized gains, and time-varying and nonnormalized gains. Interpretations of a large number of new results are given as a consequence of this analysis.(ii)Numerical problems regarding the computation of the expressions derived in [4] are explored and guidelines are given to solve them.(iii)We also provide a number of examples and discussions regarding the situations in which the expressions derived in [4] are not accurate.(iv)We propose a simple semianalytic method that makes use of the original expressions and show that the method is accurate enough for analyzing the performance of the sphericity test in fixed as well as in time-varying channels with any fading statistic. Our method is validated by simulation considering a Rayleigh fading channel as a case study. This method also corrects the one proposed in [6], where the average behavior of the fading was not correctly taken into account in the derivation of the probability of detection over a Rayleigh fading channel.(v)We analyze exemplifying situations in which different detection techniques are differently affected by the channel gains, influencing the performance ranking of these techniques or the performance gaps or both from a channel realization to another. We modify or correct accordingly the related interpretations given in [4].(vi)Last, but not least, we also give new interpretations concerning the performance of the sphericity test and other competing detectors when applied to the detection of multiple primary users. Some of these interpretations ratify those in [4], some of them contradict those provided in [4].

Along with the expressions in [4], the new results and discussions reported here constitute important tools for the understanding, the design, and the analysis of the sphericity-test-based cooperative spectrum sensing and other competing detectors over fading and nonfading channels, in the presence of multiple primary users.

*Paper Organization*. The remaining of this paper is organized as follows. In Section 2 we reproduce, in a condensed way, the main results from [4] concerning the system model and the expressions for the false alarm and the detection probabilities of the sphericity test. Section 3 presents results for validating the analytical and empirical computations throughout the paper. Section 4 is devoted to the analysis of the channel normalization and channel dynamics on the performance assessment of the sphericity test. In Section 5 our semianalytic method for computing the detection probability of the sphericity test over time-varying fading channels is described. The performance of the sphericity test and other competing detectors is investigated in Section 6, for both the fixed-gain and the time-varying fading channels. Section 7 concludes the paper summarizing the main achievements of our work.

#### 2. Main Results from [4]

In this section we describe the system model and provide the main expressions derived in [4] for computing the false alarm and the detection probabilities of the sphericity test (ST). The aim is to make this paper self-contained and facilitate the understanding and the application of such expressions. We use the same notation of [4] for the sake of consistency.

##### 2.1. System Model

The system model is the standard one, which considers a -sensor cooperative sensing in the presence of primary transmitters. The sensors may be receive antennas in one secondary receiver or single-antenna secondary devices, or any combination of these. A realization of the received data vector is where , the matrix represents the channels between the primary transmitters and the sensors, and the vector represents the zero mean transmitted signals from the primary users; denotes the transposition operation. The vector represents the complex Gaussian noise with zero mean and covariance matrix , where the noise power and the identity matrix of order . By collecting i.i.d. (independent and identically distributed) observations of the vector , the matrix is formed.

Under the assumption of constant channel matrix during the sensing interval and primary user signals following an i.i.d. zero mean Gaussian distribution and uncorrelated with the noise, the population covariance matrix of the received signal under the hypotheses of absence () and presence () of the primary signals is, respectively, where denotes complex conjugate transpose, denotes the expectation operation, and denotes the transmission power of the th primary user. In this case the th received SNR is defined by where is the Euclidian norm of the underlying vector.

Since is positive definite, that is, , the ST tests the null hypothesis against all other alternatives corresponding to . However, since the population covariance matrix is not available in practice, the sphericity test relies on the sample covariance matrix . In this case, the test statistic of the ST-based detector is where and are the determinant and the trace of , respectively, and are the ordered eigenvalues of . If the test statistic is greater than some threshold , the detector declares ; it declares otherwise.

Interestingly enough, the threshold range for the ST lies in the interval , no matter the system parameters chosen. This also differs the ST test from most of the tests for spectrum sensing, and represents a clear advantage in practice.

##### 2.2. False Alarm Probability of the Sphericity Test

From Proposition 1 in [4], for any sensor size and sample size , the two-first-moment Beta-approximation to the cumulative distribution function (CDF) of under is where is the Beta function, with being the gamma function, and is the incomplete Beta function. The parameters and are given by with where The false alarm probability as a function of the threshold is then

##### 2.3. Detection Probability of the Sphericity Test

From the Proposition 3 in [4], for any sensor size and sample size , the two-first-moment Beta-approximation to the CDF of under is where the parameters and are given by with where with being the ordered eigenvalues of .

The detection probability as a function of the threshold is

#### 3. Validation, Counterexamples, and Possible Numerical Problems

In this section we reproduce some results from [4] so that subsequent results reported here can be trusted. We also give some counterexamples in which the theoretical results from the expressions in [4] cannot be obtained due to numerical limitations, or do not match empirical results.

##### 3.1. Validation

Figures 1 and 2 show analytical and empirical results for the false alarm and the detection probability of the sphericity test as a function of the detection threshold, respectively, for some values of and . The adherence between analytical and empirical results is the same observed in [4]; one should expect possible shifts of the curves when compared with those in [4], since the realization of the channel matrix used in [4] was almost surely different from the one used to produce the corresponding results here. As in [4], we assume three primary users () with dB, dB, and dB. The entries of the channel matrix are independently drawn from a standard complex Gaussian distribution. The channel is fixed during the sensing interval and normalized as , . The power of the noise is set to , without loss of generality. Thus, the transmission power of the th primary user is computed from (3) as , and the population covariance matrices under and are formed according to (2). The empirical results were obtained from Monte Carlo simulation runs, the same number used in [4], keeping the same realization of in all runs (one simulation run corresponds to a single sensing event during which samples per sensor are collected). Thus, AWGN channels with fixed SNRs and fixed gains are considered from the primary transmitters to the sensors. The entries of the transmitted signal matrix and of the noise vector are drawn from a zero-mean complex Gaussian distribution. The entries of the th row of have variance , and the entries of the noise vector have unitary variance. In each curve there are 1000 threshold values whose minimum and maximum were, respectively, obtained from the minimum and maximum values of the test statistics under and . These values were precomputed from a separate Monte Carlo simulation with 10000 runs.